The fractional quantum Hall (FQH) states with higher Landau levels have new characters different from those with 0<ν<2. The FQH states at 2<ν<3 are examined by developing the Tao-Thouless theory. We can find a unique configuration of electrons with the minimum Coulomb energy in the Landau orbitals. Therein the electron (or hole) pairs placed in the first and second nearest Landau orbitals can transfer to all the empty (or filled) orbitals at ν0=8/3, 14/5, 7/3, 11/5, and 5/2 via the Coulomb interaction. More distant electron (or hole) pairs with the same centre position have the same total momentum. Therefore, these pairs can also transfer to all the empty (or filled) orbitals. The sum of the pair energies from these quantum transitions yields a minimum at ν=ν0. The spectrum of the pair energy takes the lowest value at ν0 and a higher value with a gap in the neighbourhood of ν0 because many transitions are forbidden at a deviated filling factor from ν0. From the theoretical result, the FQH states with ν=ν0 are stable and the plateaus appear at the specific filling factors ν0.

1. Introduction

The plateau at the filling factor ν=5/2 attracts a great deal of attention because of a new fractional quantum Hall (FQH) character. The plateau in the filling factor ν>2 has characters different from that at 0<ν<2. For example, Pan et al. [1] have found a deep minimum of the diagonal resistance, RXX and RYY, at ν=5/2 and 7/2. At ν=9/2 and 11/2 the diagonal resistance exhibits a strongly anisotropic behaviour, where RXX has a sharp peak while RYY has a minimum at ν=9/2,11/2 [1–3]. (The definition of the coordinate axes, x, y, and z, will be shown in Figure 5 of the next section.) Eisenstein et al. [4] have obtained the plateaus of Hall resistance RXY at ν=5/2 and 7/2 with even denominator. Furthermore, the other plateaus have been discovered at ν=7/3,8/3,11/5,14/5,16/5, and 19/5 with odd denominator as seen in Figure 1.

Figure 1

Behavior of Hall resistance in the region of 2<ν<4 quoted from [4].

The plateaus have the precise Hall resistance value. For example, the plateau at ν=7/2 has the Hall resistance value 2h/(7e2) within 0.015% as measured in [4]. This accuracy of the Hall resistance indicates that the ν=(7/2) state has a lower energy than the one at ν=(7/2)(1±0.00015).

Further experimental data are shown in Figure 2 which have been observed by Dean et al. [5] and Xia et al. [6]. The Hall resistance-curve in the left panel of Figure 2 [5] is different from that in the right panel [6]. This difference means that the shape of the Hall resistance versus magnetic field curve depends on the samples and the experimental conditions (magnetic field strength, etc.). In particular, the difference is large at ν=16/7,11/5.

Figure 2

Hall resistance curve in the region 2<ν<3. The left panel is quoted from [5] and the right panel from [6].

When the magnetic field is tilted from the direction perpendicular to the quasi-2D electron system, the Hall resistance plateau at ν=5/2 disappears as seen in Figure 3 which has been found by Csáthy et al. [7]. On the other hand, the ν=7/3,8/3 plateaus persist with the tilt as in Figure 3.

Figure 3

Tilt dependence of the Hall resistance and diagonal resistance [7].

The temperature dependence of RXX has been measured by many researchers. For example, the temperature dependence of the diagonal resistance curves has been measured by Pan et al. [8]. The diagonal resistance curve at 36 mK is different from that at 6 mK. Furthermore, some local minima in the diagonal resistance curve disappear at 36 mK. Using the temperature dependence of RXX, the Arrhenius plots are drawn to give an energy gap. The energy gap is shown in Figure 4 which is obtained by Choi et al. [9] and also in the paper [5].

Figure 4

Energy gaps for the FQH states. Open circles are quoted from [5]. Solid circles and squares are quoted from [9].

Figure 5

Quantum Hall Device.

The data obtained in the high mobility sample [9] give the energy gap for the filling factors of 14/5,19/7,8/3,5/2,7/3,16/7, and 11/5 as in Figure 4.

These experimental findings at ν>2 have stimulated theoretical studies. Several theories have been proposed to explain the plateaus of the Hall resistance at ν>2, especially at ν=5/2. Some of them are briefly reviewed below.

Koulakov et al. have studied the ground state of a partially filled upper Landau level in a weak magnetic field. They have used the effective interaction [10] which was derived by Aleiner and Glazman in the 2D-electron system with high Landau levels, taking into account the screening effect by the lower fully occupied levels. Then, they have found that the ground state is a charge-density wave (CDW) state with a large period [11]. Moessner and Chalker studied a 2D-electron system with a fermion hardcore interaction and without disorder. They found a transition to both unidirectional and triangular charge-density wave states at finite temperatures [12]. Rezayi et al. numerically studied a 2D-electron system in magnetic field with a high Landau level half filled by electrons. In finite size systems with up to 12 electrons and torus geometry, they found a charge-density wave ordering in the ground state. Their results show that the highest weight single Slater determinant has the occupation pattern 11111000001111100000, where 1 and 0 stand, respectively, for an occupied orbital and an empty orbital [13].

Haldane and Rezayi investigated the pair state with spin-singlet [14]. They used a hollow core Hamiltonian. In the Landau level number L=1, the hollow core Hamiltonian has the first pseudopotential V1>0 although the zeroth Haldane pseudopotential V0 is zero. They found a ground state called HR state. Moore and Read were inspired by the structure of the HR state, and constructed the pair state a p-wave (px-ipy) polarized state. They have described the FQH state in terms of conformal-field-theory [15]. The state is called the Moore-Read state (MR state). In [16], Read wrote “the wavefunction ψMR represents BCS [17, 18] pairing of composite fermions. One type are the charged vortices discussed above, with charge 1/(2q) which according to MR are supposed to obey nonabelian statistics.” Greiter et al. investigated the MR state from the viewpoint of the composite fermion pair [19, 20]. The statistics are an ordinary abelian fractional statistics.

Morf argued the quantum Hall states at ν=5/2 by a numerical diagonalization [21]. He studied spin-polarized and -unpolarized states with N≤18 electrons. His result indicates that the 5/2 state is expected to be the spin-polarized MR state. Rezayi and Haldane [22] confirmed Morf’s results. Their results are based on numerical studies for up to 16 electrons in two geometries: sphere and torus. They found a first order phase transition from a striped state to a strongly paired state. They examined 12 electrons in a rectangular unit cell with the aspect ratio 0.5. They found the stripe state, the probability weight of which is 58% for the single Slater determinant state with the occupation pattern 000011110000111100001111. Also, they found an evidence that the ν=5/2 state is derived from a paired state which is closely related to the MR polarized state or, more precisely, to the state obtained particle-hole (PH) symmetrisation of the MR state [22].

Tao and Thouless [23, 24] investigated the FQH states in which the Landau states with the lowest energy are partially filled with electrons. Thus, the theory does not assume any quasiparticle. The present author has developed a theory on the FQH states at 0<ν<2 [25–32] by extending the Tao-Thouless theory. We will apply the theory to the problem of 5/2 plateau in Section 3. The plateaus at 2.5<ν<3 and at 2<ν<2.5 are discussed in Sections 5 and 6, respectively. Before examining this problem, the fundamental properties will be shortly summarized for the quasi-2D electron system below.

2. The Properties of a Quasi-2D Electron System

A quantum Hall device is illustrated in Figure 5 where the x-axis is the direction of the current and the y-axis is the direction of the Hall voltage. Then, the vector potential, A, has the components
(1)A=(-yB,0,0),
where B is the strength of the magnetic field. The Hamiltonian, H0, of a single electron in the absence of the Coulomb interaction between electrons is given by
(2)H0=(p+eA)22m*+U(y)+W(z),
where U(y) and W(z) indicate the potentials confining electrons to an ultrathin conducting layer in Figure 5. Therein m* is an effective mass of electron and p=(px,py,pz) is the electron momentum. The Landau wave function of the single electron is given by (3a)ψL,J(x,y,z)=1ℓexp⁡(ipxℏ)uLHL(m*ωℏ(y-αJ))×exp⁡(-m*ω2ℏ(y-αJ)2)ϕ(z),(3b)ω=eBm*, where ϕ(z) is the wave function of the ground state along the z-direction, HL is the Hermite polynomial of Lth degree, uL is the normalization constant, and ℓ is the length of the quasi-2D electron system as in Figure 5. The integer L is called Landau level number hereafter.

Because of the periodic boundary condition, the momentum p is given by
(4)p=[2πℏℓ]×J.
The momentum is related to the value αJ in the wave function, (3a), as
(5)αJ=p(eB)=[2πℏ(ℓeB)]J.
The eigenenergy is given by
(6)EL,J=λ+U(αJ)+(ℏeBm*)(L+12)(L=0,1,2,3,…),
where L is the Landau level number, λ is the ground state energy along the z-direction, and U(αJ) is the potential energy in the y-direction.

When there are many electrons, the total Hamiltonian is given by
(7)HT=∑i=1NH0,i+∑i=1N-1∑j>iNe24πε(xi-xj)2+(yi-yj)2+(zi-zj)2,
where N is the total number of electrons, ε is the permittivity and H0,i is the single particle Hamiltonian of the ith electron without the Coulomb interaction as
(8)H0,i=(pi+eA)22m*+U(yi)+W(zi).
The many-electron state is characterized by a set of Landau level numbers L1,L2,…,LN and a set of momenta p1,p2,…,pN. The complete set is composed of the Slater determinant as
(9)Ψ(L1,…,LN;p1,…,pN)=1N!|ψL1,p1(x1,y1,z1)⋯ψL1,p1(xN,yN,zN)⋮⋮ψLN,pN(x1,y1,z1)⋯ψLN,pN(xN,yN,zN)|.
This state is the eigenstate of ∑i=1NH0,i. The expectation value of the total Hamiltonian is denoted by W(L1,…,LN;p1,…,pN) which is given by
(10)W(L1,…,LN;p1,…,pN)=∑i=1NELi(pi)+C(L1,…,LN;p1,…,pN),
where C is the expectation value of the Coulomb interaction defined by
(11)C(L1,…,LN;p1,…,pN)=∫⋯∫Ψ(L1,…,LN;p1,…,pN)*×∑i=1N-1∑j>iNe24πε(xi-xj)2+(yi-yj)2+(zi-zj)2×Ψ(L1,…,LN;p1,…,pN)dx1dy1dz1⋯dxNdyNdzN.
Hereafter, we call C(L1,…,LN;p1,…,pN) “classical Coulomb energy.” We divide the total Hamiltonian HT into two parts HD and HI as follows: (12a)HD=∑L1,…,LN∑p1,…,pN|Ψ(L1,…,LN;p1,…,pN)〉×W(L1,…,LN;p1,…,pN)×〈Ψ(L1,…,LN;p1,…,pN)|,(12b)HI=HT-HD, where HI is composed of the off-diagonal elements only. Accordingly, the total Hamiltonian HT of the quasi-2D electron system is a sum of HD and HI as follows:
(13)HT=HD+HI.
The Slater determinant composed of the Landau states is the exact eigenstate of HD. So we will examine the residual part, namely, quantum transitions via the off-diagonal parts of the Coulomb interaction.

Because the Coulomb interaction depends only upon the relative coordinate of electrons, the total momentum along the x-direction conserves in the quasi-2D electron system. That is to say, the sum of the initial momenta pi and pj is equal to that of the final momenta pi′ and pj′:
(14)pi′+pj′=pi+pj.

Next we discuss the configuration of electrons in the Landau orbitals. The previous article [32] has verified that the most uniform configuration of electrons is uniquely determined for any filling factor except at both ends. The effects of the boundaries may be neglected in a macroscopic system. At ν<1 the Landau states with the Landau level number L=0 are partially occupied by electrons and all the states with L≥1 are empty. For example, the most uniform configuration for ν=2/3 is constructed by repeating of the unit configuration (filled, empty, filled) as shown in Figure 6. This configuration determines the set of the momenta as (15a)p2n-1=p1+(2πℏℓ)×3(n-1),n=1,2,3,…,(15b)p2n=p1+(4πℏℓ)+(2πℏℓ)×3(n-1) for the filling factor23, where p1 is the minimum value of the momentum.

Figure 6

Most uniform configuration of electrons at ν=2/3. The current flows along the x-axis and the Hall voltage yields along the y-axis. Red solid lines indicate the Landau orbitals filled with electron. Blue dashed lines indicate the empty orbitals.

For an arbitrary filling factor, we can also find the most uniform configuration of electrons in the Landau states. Then, the configuration yields the minimum expectation value of HD, namely, the ground state of HD.

We next count the number of the Coulomb transitions via HI. When the filling factor deviates a little from the specific filling factor, the number of quantum transitions decreases abruptly because of the Fermi-Dirac statistics and the momentum conservation. That is to say, the number of the Coulomb transitions at the specific filling factors takes the largest among those of the neighbouring filling factors. This property produces the minimum energy at the specific filling factors and yields the precise confinement of the Hall resistance. This mechanism can explain the phenomena of the FQHE at ν<2 without introducing any quasiparticles [25–32].

Here, we remark the edge current in the FQH states. Büttiker [33] investigated the current distribution in a 2D-electron system and found the edge current. Both total current and Hall voltage are affected by the edge current in the IQHE but the Hall resistance remains to be the original value. The mechanism has been studied for the FQH states under the existence of the edge current in the article [29]. The precise confinement of the Hall resistance is derived from the momentum conservation along the current direction.

3. Explanation for the Appearance of 5/2 Plateau

We first compare the energy gap at ν>2 with that at ν<2. The energy gap at ν=5/2,8/3,7/3, and so forth is determined from the experiment [9], the results of which are listed in Table 1.

Table 1

Energy gap measured for the fractional quantum Hall states in the second Landau level [9].

ν

ν = 14/5

ν = 19/7

ν = 8/3

ν = 5/2

ν = 7/3

ν = 16/7

ν = 11/5

Sample A

252 mK

108 mK

562 mK

544 mK

584 mK

94 mK

160 mK

Sample B

<60 mK

150 mK

272 mK

206 mK

<40 mK

The energy gap for ν<1 is shown in Figure 7 which is obtained in [34]. The value of the energy gap changes from sample to sample as in Table 1. The energy gap at ν=2/3 is about 4.3 K and that of ν=5/2 is about 0.272~0.544 K as measured in [9] and [34]. Thus, the energy gap in the region 2<ν<3 is about 1/10 times that in ν<1. Therefore, we cannot ignore the small terms of various Coulomb transitions in studying the energy spectrum for ν>2.

Figure 7

Energy gaps at ν=2/3,3/5,4/7,5/9,6/11,5/11,4/9,3/7,2/5 in [34].

In the region of 2<ν<3, all the Landau states with L=0 are filled with electrons with up and down spins, and the Landau states with L=1 are partially occupied by electrons. The interactions between electrons depend on the shape of the Landau wave function in the x and y directions, the wave function along the z-direction, the screening effect of electrons in the lowest Landau level, and so on. The theories [10–22] reviewed in Section 1 have employed various types of interactions. For example, the first Haldane pseudopotential V1 for L=1 is positive although V0 for the lowest Landau level (L=0) is zero. Thus, the wave function and the interaction in higher Landau levels are different from those in the lowest Landau level.

In the previous articles [25–32], we have ignored the energy from the pairs placed in the second nearest neighboring Landau orbitals, because it is expected to be smaller than that in the first nearest Landau orbitals. For the ν>2 FQH states we have to include the contribution from the electron pairs placed in the second neighboring Landau orbitals because of the small energy-gap as in Table 1.

We first study the ν=5/2 FQH state using the method of the previous papers [25–32]. At ν=5/2=2+1/2, the most uniform configuration of electrons in the L=1 level is illustrated in Figure 8 where the fully occupied orbitals with L=0 are not drawn for simplicity.

We examine the quantum transitions via the Coulomb interaction HI. All the Coulomb transitions satisfy the momentum conservation along the x-axis. Figure 8 shows schematically the quantum transitions from the electron pair CD as an example. The momenta of electrons at C and D are described by pC and pD, respectively. These momenta change to pC′ and pD′ after the transition. The momentum conservation gives the following relation: (16a)pC′=pC-Δp,(16b)pD′=pD+Δp, where Δp is the momentum transfer. The quantum transition is allowed to empty orbitals only. As seen in Figure 8, the empty orbital exists in the odd numbered orbitals from the left of the orbital C. Therefore, the transfer momentum takes the following value derived from (4) and (5):
(17)Δp=(2πℏℓ)×(2n-1)n=±1,±2,±3,±4,…,
where n=0 is eliminated because the transition is forbidden by the Pauli exclusion principle. All the allowed transitions are illustrated by the green arrow pairs in Figure 8. Thus, any electron pair placed in the second neighboring orbitals can also transfer to all the empty orbitals (except n=0) at ν=5/2.

In order to calculate the pair energies, the following summation S is introduced for the Landau level L=1: (18a)S=-∑Δp≠0,-4πℏ/ℓ〈L=1,pC,pD|HI|L=1,pC′,pD′〉×〈L=1,pC′,pD′|HI|L=1,pC,pD〉WG-Wexcite(pC⟶pC′,pD⟶pD′),(18b)pD=pC+4πℏℓ,(18c)pC′=pC-Δp,pD′=pD+Δp. The summation is carried out for all the momentum changes Δp=(2πℏ/ℓ)×integer except Δp=0 and -4πℏ/ℓ. The elimination comes from disappearance of the diagonal matrix element of HI. The summation S is positive, because the denominator in (18a) is negative. The perturbation energy ςCD of the pair CD is expressed by the summation S as follows:
(19)ςCD=-(12)S,
because the function in (18a) is continuous for the argument Δp and also the momentum change 2πℏ/ℓ is extremely small for a macroscopic size of the device. Therein the factor 1/2 comes from the fact that the number of allowed transitions is equal to the number of the empty orbitals which is half of the total Landau orbitals with L=1.

There are many electron pairs like CD. The total number of the pairs like CD is equal to Nν=5/2L=1 which indicates the total number of electrons placed in the Landau orbitals with L=1. Accordingly, the perturbation energy of all the second nearest electron pairs is given by
(20)Eν=5/2pair=-(12)SNν=5/2L=1.
The pair energy per electron is
(21)Eν=5/2pairNν=5/2L=1=-(12)S.
The summation S depends on the thickness, size, and material of the quasi-2D electron system. The reasons are as follows. The wave function along the z-axis depends on both thickness and potential shape along the z-axis. The wave function length of the x-direction depends on the device size. The effective mass of electron and the permittivity depend on the material of the device. Therefore, the classical Coulomb energy W and the transition matrix element vary with changing the quantum Hall device. Accordingly, the value of S varies from sample to sample.

Furthermore, the L=1 Landau wave function is zero at its center position because of the Hermite polynomial of L=1 degree as in (3a). Accordingly, the function form in (18a) is quite different from that for L=0. Additionally, we need to consider the screening effect from the L=0 electrons. The effect is also unknown. Therefore, we do not go into detail of the summation and treat S as a parameter.

We have ignored the quantum transitions into higher Landau levels with L≥2. The contribution is extremely small because the excitation energy is very large as follows: the excitation energy from the Landau level with L to that with L+1 is given by
(22)EL+1-EL=ℏeBm*,
which is derived from (6). The effective mass m* differs from material to material and the value in GaAs is about 0.067 times that of free electron. For example, this excitation energy is estimated at the magnetic field strength 4T as
(23)E2-E1≈1.055×10-34×1.602×10-19×4(0.067×9.109×10-31)≈1.108×10-21[J],(E2-E1)kB≈80.3[K]forB=4T.
In the perturbation calculation, the denominator is the energy difference WG-Wexcite. When the intermediate state belongs to L=2, the main part of WG-Wexcite is E1-E2 for 2<ν<4. The value (E2-E1)/kB is about 80[K], and so the absolute of the denominator for the intermediate states with L=2 is very large compared with that for the intermediate states with L=1 in (18a). Therefore, the contribution from the intermediate states with L≥2 is extremely small, so we ignore them. We examine now the energy gaps in Table 1 which have the magnitude of about 0.1 K. The absolute value of the denominator for the intermediate state with L=2 is about 800 times the energy gaps. Accordingly, the intermediate states with L≥2 may be neglected.

Next, we study the perturbation energy in the neighborhood of ν=5/2. As an example, the ν=48/19=2+(10/19) state is examined. The most uniform electron configuration is illustrated in Figure 9 where the Landau orbitals with L=0 are not shown, for simplicity. The unit configuration is (1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0), where 1 indicates a filled orbital and 0 an empty orbital.

Figure 9

Quantum transitions at ν=48/19 state.

The electron pair CD can transfer to all the empty orbitals as shown by green arrow pairs. On the other hand the pair GA can transfer to only one site per unit configuration as shown by black arrow pairs in Figure 9 where the red symbol × indicates the forbidden transition. So the pair energy ςGA′ is
(24)ςGA′=-(119)Sforν=4819.
Therein the coefficient 1/19 indicates that the electron pair GA can transfer to one site per 19 Landau orbitals. The energies of the other pairs are calculated, the results of which are
(25)ςFG′=-(319)S,ςEF′=-(519)S,ςDE′=-(719)S,ςCD′=-(919)S,forν=4819.
We calculate the electron pair AB which is placed in the nearest neighboring Landau orbitals as in Figure 9. In order to calculate the energy of the nearest pair, we introduce the following summation T:
(26)T=-∑Δp≠0,-2πℏ/ℓ〈L=1,pA,pB|HI|L=1,pA′,pB′〉×〈L=1,pA′,pB′|HI|L=1,pA,pB〉WG-Wexcite(pA⟶pA′,pB⟶pB′),
where
(27)pB=pA+2πℏℓ,
which is different from (18b). This summation T is positive because the denominator of (26) is negative. The value T is also treated as a parameter because T varies from sample to sample. The pair AB can transfer to any empty state as shown by the red arrow pairs in Figure 9. The number of the empty states is 9 per unit configuration. Accordingly, the pair energy of AB is given by
(28)ςAB′=-(919)T,forν=4819.
The total perturbation energy from all the electron pairs placed in the first and second neighboring Landau orbitals with L=1 is
(29)Eν=48/19pair=(ζAB′+ζCD′+2ζDE′+2ζEF′+2ζFG′+2ζGA′)(Nν=48/19L=110).
Substitution of (24), (25), and (28) into (29) yields
(30)Eν=48/19pair=-((919)T+(4119)S)·(Nν=48/19L=110).
The pair energy per electron for L=1 is
(31)Eν=48/19pairNν=48/19L=1=-((9/19)T+(41/19)S)10.
One more example ν=78/31=2+(16/31) will be examined. The most uniform configuration is shown in Figure 10 where the Landau orbitals with L=0 are not shown. The electron pair CD can transfer to all the empty orbitals as shown by green arrows. On the other hand, the pair JA can transfer to only one site per unit configuration as illustrated by black arrows in Figure 10.

Figure 10

Quantum transitions at ν=78/31 state.

Accordingly, the pair energy ςJA′′ is
(32)ςJA′′=-(131)Sforν=7831.
The perturbation energies of the other second nearest pairs are given by
(33)ςIJ′′=-(331)S,ςHI′′=-(531)S,ςGH′′=-(731)S,ςFG′′=-(931)S,ςEF′′=-(1131)S,ςDE′′=-(1331)S,ςCD′′=-(1531)S,forν=7831.
The electron pair AB can transfer to any empty state as shown by the red arrows in Figure 10. The number of the empty states is 15 per unit configuration. Accordingly, the pair energy of AB is given by
(34)ςAB′′=-(1531)Tforν=7831.
The total perturbation energy from all the electron pairs placed in the first and second neighboring Landau orbitals with L=1 is
(35)Eν=78/31pair=(ζAB′′+ζCD′′+2ζDE′′+2ζEF′′+2ζFG′′+2ζGH′′+2ζHI′′+2ζIJ′′+2ζJA′′)×(Nν=78/31L=116).
Substitution of (32), (33), and (34) into (35) yields
(36)Eν=78/31pair=-((1531)T+(11331)S)·(Nν=78/31L=116).
The pair energy per electron for L=1 is
(37)Eν=78/31pairNν=78/31L=1=-((15/31)T+(113/31)S)16forν=7831.
This filling factor ν=78/31=2.5161⋯ is close to ν=5/2. The difference between ν=5/2 and 78/31 is about 0.6%. We compare the perturbation energy of the first and second nearest pairs per electron between ν=5/2 and 78/31:
(38)Eν=5/2pairNν=5/2L=1-Eν=78/31pairNν=78/31L=1=(T-9S)(15496).
From (38) the pair energy at ν=5/2 is lower (or higher) than that at ν=78/31 for T-9S<0 (or T-9S>0). We examine the following two Cases 1 and 2.

Case 1 (9S≫T).

In this case, the perturbation energy of the first and second pairs per electron is
(39)Eν=5/2pairNν=5/2L=1≪Eν=78/31pairNν=78/31L=1.
Accordingly the pair energy at ν=5/2 is sufficiently lower than that in the neighborhood of ν=5/2. So the ν=5/2 state is very stable and the Hall plateau appears at ν=5/2.

Case 2 (9S≪T).

In this case, the ν=5/2 Hall plateau does not appear because the pair energy at ν=5/2 is higher than that in its neighborhood.

Thus the FQH state is sensitive to the relative value of S and T which is dependent on the materials, thickness of the conducting layer, device structure, and so on. In the next section, we discuss the sample dependent phenomena based on the theory obtained above.

4. Sample Dependent Phenomena

For example, the 5/2 and 7/2 Hall plateaus do not exist on the red curve of Hall conductance obtained by Dean et al. in the article [35] as seen in Figure 11. On the other hand, the experimental results in Figures 1–3 indicate the appearance of the 5/2 and 7/2 Hall plateaus. Thus, the appearance or disappearance of the ν=5/2 and 7/2 plateaus seems to depend upon the samples used in the experiments.

Figure 11

Plateaus of Hall conductance and local minima of diagonal resistance in the experimental results of [35].

(Note: we point out similar phenomena at ν=1/2. The Hall plateau appears at ν=1/2 in the experimental results [36–40] but disappears in the ordinal experiments for example in the article [34].)

According to our theory examined in Section 3, this property comes from the relative value of S and T.

5. FQH States at Filling Factors 2.5<ν<3

As will be shown below, in the region of 2.5<ν<3, we find fractional filling factors where both first and second nearest electron pairs can transfer to all the empty Landau orbitals with L=1. These filling factors are ν= 8/3, 14/5, 18/7, and 19/7 for the denominator smaller than nine. The allowed transitions are shown by the arrow pairs in Figures 12, 13, 14, and 15.

Figure 12

Quantum transitions at ν=8/3 state.

Figure 13

Quantum transitions at ν=14/5 state.

Figure 14

Quantum transitions at ν=18/7 state.

Figure 15

Quantum transitions at ν=19/7 state.

Figure 12 shows the most uniform configuration at ν=8/3=2+(2/3) where two types of electron pairs exist. The pair AB represents the first nearest electron pair and the pair CD the second nearest one. Both the pairs AB and CD can transfer to all the empty orbitals with L=1. The allowed transitions are shown by black and green arrow pairs, respectively, in Figure 12.

The number of empty orbitals with L=1 is 1/3 of the Landau orbitals with L=1. Therefore, the pair energies are given by (40a)ςAB=-(13)Tforν=83,(40b)ςCD=-(13)Sforν=83.

The total energy of the electron pairs placed in the first and second neighboring Landau orbitals with L=1 is (41a)Eν=8/3pair=ςAB×(Nν=8/3L=12)+ςCD×(Nν=8/3L=12),
where Nν=8/3L=1 indicates the total number of electrons in the Landau orbitals with L=1. Substitution of (40a) and (40b) into (41a) yields
(41b)Eν=8/3pair=-(T6+S6)Nν=8/3L=1. We examine the pair energy in the limit from the right or left to ν=8/3. Using the same method reported in the previous papers [25–32], we obtain the right and left hand limits as (42a)lim⁡ν→(8/3)+εEνpair=-(T12+S12)Nν=8/3L=1,(42b)lim⁡ν→(8/3)-εEνpair=-(T12+S12)Nν=8/3L=1. Therefore, a valley in the energy spectrum appears as
(43)ΔEν=8/3pair=Eν=8/3pair-lim⁡ν→(8/3)±εEνpair=-112(T+S)Nν=8/3L=1.
The ν=14/5,18/7, and 19/7 states have the most uniform configuration as shown in Figures 16, 17, and 18, respectively. The allowed transitions are schematically drawn by the black and green arrow pairs for the first and second nearest electron pairs, respectively.

Various electron pairs with the same total momentum at ν=8/3. Dashed lines indicate empty orbitals and solid lines indicate filled orbitals in the Landau level L=1. Allowed transitions from the electrons J and K are shown by black arrow pairs, from IL by blue, from HM by brown, and from GN by dark green.

The number of the allowed transitions is 1/5 times the number of Landau orbitals with L=1. Then, the pair energy of AB and CD is given, respectively, by (44a)ςAB=-(15)Tforν=145,(44b)ςCD=-(15)Sforν=145. There are many electron pairs represented by AB and CD. The total number of the pairs represented by AB is equal to (1/4)Nν=14/5L=1 and also that by CD is equal to (1/4)Nν=14/5L=1. Then, we obtain the total pair energy for the electron pairs placed in the first and second neighboring Landau orbitals with L=1 as
(45)Eν=14/5pair=-(T20+S20)Nν=14/5L=1.
Figure 14 shows the allowed transitions of the pairs AB and CD at the filling factor ν=18/7. The number of the empty orbitals is 3/7 times that of the Landau orbitals with L=1. Accordingly, the pair energy of AB and CD is given, respectively, by(46a)ςAB=-(37)Tforν=187,(46b)ςCD=-(37)Sforν=187. Then we obtain
(47)Eν=18/7pair=-(3T28+3S28)Nν=18/7L=1.
Next we count the number of allowed transitions of the pairs AB and CD at ν=19/7. The electron pairs AB and CD in Figure 15 can transfer to all the empty Landau orbitals with L=1.

Since the number of the allowed transitions for each of the AB and CD pairs is two per unit configuration composed of the seven Landau orbitals, the pair energy of AB and CD is given, respectively, by (48a)ςAB=-(27)Tforν=197,(48b)ςCD=-(27)Sforν=197. Then we obtain
(49)Eν=19/7pair=-(2T35+2S35)Nν=19/7L=1.
Thus, the electron pairs AB and CD can transfer to all the empty orbitals at ν= 5/2, 8/3, 14/5, 18/7, and 19/7, and therefore the pair energy becomes very low, resulting in a strong binding energy.

The values of S and T may vary from sample to sample. We examine the condition that the 5/2 plateau appears. From (38), the ν=5/2 state is stable when S is sufficiently lager than T/9. In the experiment [9], the energy gaps have been measured as in Figure 4. In the high mobility sample [9], the energy gap at ν=5/2 is nearly equal to that at ν=8/3. Equations (21) and (41b) give the pair energy per electron as follows: (50a)Eν=5/2pairNν=5/2L=1=-(S2),(50b)Eν=8/3pairNν=8/3L=1=-(T6+S6). The experimental data [9] can be explained by the present theory under the following condition:
(51)T≈2S.
Equations (21), (41b), (45), and (49) give the theoretical ratio of the pair energies at ν=5/2,8/3,14/5, and 19/7 as follows:
(52)|Eν=5/2pairNν=5/2L=1|:|Eν=8/3pairNν=8/3L=1|:|Eν=14/5pairNν=14/5L=1|:|Eν=19/7pairNν=19/7L=1|=(S2):(T6+S6):(T20+S20):(2T35+2S35).
When condition (51) is satisfied, the theoretical ratio of the pair energies becomes
(53)(S2):(T6+S6):(T20+S20):(2T35+2S35)=1:1:(310):(1235).
From Figure 4, the experimental data of the energy gap yield the ratio for the high mobility sample as
(54)0.0047:0.005:0.0023:0.001=0.94:1:0.46:0.2.
Thus, the present theory explains reasonably well the experimental data.

6. FQH States at Filling Factors 2<ν<2.5

Next we examine the FQH states with 2<ν<2.5. The most uniform configurations at ν=7/3,11/5,17/7, and 16/7 are schematically drawn in Figure 16. The hole-pairs AB and CD can transfer to all the electron states in L=1 as easily seen in Figure 16. This property produces a strong binding energy between the hole-pairs.

The number of allowed transitions for the hole-pairs at ν=7/3 is equal to that for the electron-pairs at ν=8/3. This symmetry between electron and hole is clearly seen by comparing the number of transitions (namely, number of arrows) for ν= 8/3 and 7/3, as easily seen in Figure 17.

From the discussion given in Sections 3, 5, and 6, we find that the state with ν=5/2,7/3,8/3,11/5,14/5,16/7,17/7, and 18/7 and 19/7 is stable in the region 2<ν<3. The left panel of Figure 2 quoted from [5] shows the plateaus of the Hall resistance at ν=5/2,7/3,8/3,11/5,14/5, and 16/7. Similar investigation for ν>3 can be performed by using the method of this section. The pair energy becomes large at the filling factor ν=7/2,10/3,11/3,16/5, and so on. The Hall plateaus at these filling factors have been found in several experiments.

The number of the allowed transitions via the Coulomb interaction discontinuously varies with changing the filling factor. The discontinuous variation is caused by the combined effect of the momentum conservation along the current, the most uniform configuration of electrons and the Fermi-Dirac statistics. This effect produces the stability of FQH states at the several filling factors.

As described in Section 1, different states have been proposed by different authors. For example, the ν=5/2 FQH state is explained by the stripe HR or MR states and so on. The ν=7/3 FQH state is said to be composed of the composite fermions where each electron binds to two flax quanta and the ν=11/5 FQH state is explained to be composed of the composite fermions where each electron binds to four flax quanta. The theory presented here explains the FQH phenomena occurring at ν<2 and those at 2<ν<3 in a coherent way without assuming any quasi-particles.

7. Further Investigation of the Pair Energy for 2<ν<3

We examine the exact energy of any FQH state. The total energy ET of the quasi-2D electron system is the sum of the eigenenergy W of HD and the pair energy Epair via the interaction HI as follows:
(55)ET=W+Epair,
where W has been already given by (10) as follows:
(56)W=∑i=1NELi(pi)+C(L1,…,LN;p1,…,pN).
Equations (5) and (6) yield the following equation:
(57)W=C(L1,…,LN;p1,…,pN)+Nλ+∑i=1NU(pi(eB))+∑i=1N(ℏeBm*)(Li+12).
The energy difference between different Landau levels is extremely large as shown in (22) and (23). The interval of Landau energies has been estimated for GaAs
(58)(E1-E0)kB=(E2-E1)kB≈80.3[K]forB=4T.
The experimental values of the energy gaps at 2<ν<3 are smaller than 1 [K] as in Table 1 and therefore higher Landau levels can be ignored at low temperatures. So, the ground state at 2<ν<3 is obtained by superposing many-electron states that all the Landau states with L=0 are occupied by electrons with up and down spins and the Landau states with L=1 are partially occupied by electrons. We express the number of electrons in the Landau level L by NνL and the number of Landau orbitals by NlevelL, respectively. The ratio NνL/NlevelL is described as follows: (59a)NνL=0NlevelL=2in the ground state with2<ν<3,(59b)NνL=1NlevelL=ν-2in the ground state with2<ν<3,(59c)NνL=2NlevelL=0in the ground state with2<ν<3,(59d)NνL>2NlevelL=0in the ground state with2<ν<3, where NlevelL depends on the sample and the magnetic field strength but is independent of L. The total number N of electrons is
(60)N=NνL=0+NνL=1in the ground state with2<ν<3.
Substitution of (59a), (59b), (59c), (59d), and (60) into (57) gives the eigenenergy W of HD as follows:
(61)W=C(L1,…,LN;p1,…,pN)+Nλ+∑i=1NU(pi(eB))+12(ℏeBm*)NνL=0+32(ℏeBm*)NνL=1in the ground state with2<ν<3.
Next, we investigate the pair energy which is caused by the quantum transitions via HI. The electron pairs in the ground state with 2<ν<3 have been classified into the following three types:

First type: both electrons in the pair are placed in the orbitals with L=0 only.

Second type: one electron is placed in L=0 and the other in L=1.

Third type: both electrons in the pair are placed in L=1 only.

These pair energies are described by the symbols EL=0pair, EL=0and1pair, and EL=1pair, respectively. The total energy of all the electron pairs is
(62)Epair=EL=0pair+EL=0and1pair+EL=1pairin the ground state with2<ν<3.
Therein the pair energies EL=0pair and EL=0and1pair are negligibly small because of the following reason. Any order of the perturbation energy is obtained by a summation of the functions with the denominator containing the energy difference of W between the ground and intermediate states. Any electron pair belonging to the first or second types can transfer only to the intermediate states with a higher Landau level because all the Landau orbitals with L=0 are already occupied by electrons with up and down spins. Therefore, the energy difference between the ground and intermediate states is very large as in (58). Then we may ignore the pair energy belonging to the first and second types:
(63)EL=0pair≈0,EL=0and1pair≈0in the ground state withν>2.
On the other hand, the electron pairs in L=1 can transfer to empty orbitals with L=1 for 2<ν<3. Then, the energy difference of W between the ground and the intermediate states comes from the difference in the classical Coulomb energies and so the difference is very small. We will examine any electron (or hole) pair placed in any Landau orbitals with L=1. As an example, we discuss the case of ν=8/3. Figure 18 schematically shows the electron pairs at ν=8/3. The electron pairs IL, HM, and GN possess the total momentum same as that of the pair JK. These pairs can transfer to all the empty orbitals as easily seen in Figure 18.

The momenta of the electrons at G, H, I, J, K, L, M, and N are described by the symbols pG,pH,pI,pJ,pK,pL,pM, and pN, respectively. Then, the total momenta of the electron pairs take the same value because of (4) and (5):
(64)ptotal=pG+pN=pH+pM=pI+pL=pJ+pK.
The energies of the pairs GN, HM, IL, and JK, ςGN,ςHM,ςIL, and ςJK, can be reexpressed systematically by using a symbol ςνL=1(ptotal,j), where ptotal and j indicate the total momentum and the distance between the pair as follows: (65a)ςJK=ςν=8/3L=1(ptotal,j=1),(65b)ςIL=ςν=8/3L=1(ptotal,j=5),(65c)ςHM=ςν=8/3L=1(ptotal,j=7),(65d)ςGN=ςν=8/3L=1(ptotal,j=11). Therein the momentum of each electron is given as (66a)pJ=12(ptotal-1×2πℏℓ),pK=12(ptotal+1×2πℏℓ),(66b)pI=12(ptotal-5×2πℏℓ),pL=12(ptotal+5×2πℏℓ),(66c)pH=12(ptotal-7×2πℏℓ),pM=12(ptotal+7×2πℏℓ),(66d)pG=12(ptotal-11×2πℏℓ),pN=12(ptotal+11×2πℏℓ). Thus, any momentum-pair (pV,pW) is related to ptotal and j as (67a)pV=12(ptotal-j×2πℏℓ),pW=12(ptotal+j×2πℏℓ).(67b)ptotal=pV+pW,j=(pW-pV)(2πℏ/ℓ). Because both momenta pV and pW should be equal to (2πℏ/l)×integer, the values of ptotal and j are classified to the following two cases: (68a)ptotal=(2πℏℓ)×(oddinteger)forj=(oddinteger),(68b)ptotal=(2πℏℓ)×(eveninteger)forj=(eveninteger). We have already examined the case of odd integer j in Figure 18.

Next, we examine the case of even integer j. Figure 19 shows quantum transitions with even integers j given by (68b). All the electron pairs possessing the total momentum same as that of the pair KL can transfer to all the empty orbitals as in Figure 19. The electron pairs KL, JM, IN, and HO indicate the cases of j=2,4,8, and 10, respectively.

Figure 19

Various electron pairs with the same total momentum at ν=8/3. Dashed lines indicate empty orbitals and solid lines indicate filled orbitals in the Landau level L=1. Allowed transitions from the electrons K and L are shown by black arrow pairs, from JM by blue, from IN by brown, and from HO by dark green.

The pair energies are described as (69a)ςKL=ςν=8/3L=1(ptotal,j=2),(69b)ςJM=ςν=8/3L=1(ptotal,j=4),(69c)ςIN=ςν=8/3L=1(ptotal,j=8),(69d)ςHO=ςν=8/3L=1(ptotal,j=10). The total energy of all the electron pairs is described by the symbol Epair defined by (55). Use of (62) and (63) gives
(70)Epair≈EL=1pair(ν)in the ground state with2<ν<3.
This energy at L=1 is the sum of all the pair energies with ptotal and j:
(71)EL=1pair(ν)=∑ptotal,jςνL=1(ptotal,j)in the ground state with2<ν<3.
Equations (55), (70), and (71) yield the total energy of the quasi-2D electron system as follows:
(72)ET≈W+∑ptotal,jςνL=1(ptotal,j)in the ground state with2<ν<3.
Substitution of (61) into (72) gives
(73)ET≈C(L1,…,LN;p1,…,pN)+Nλ+∑i=1NU(pi(eB))+12(ℏeBm*)NνL=0+32(ℏeBm*)NνL=1+∑ptotal,jςνL=1(ptotal,j)inthegroundstatewith2<ν<3.
We express the pair energy per electron by the symbol ξνL=1(j) which is defined by
(74)ξνL=1(j)=∑ptotalςνL=1(ptotal,j)NνL=1in the ground state with2<ν<3.
The exact pair energy is the sum of all order terms in the perturbation calculation as follows:
(75)ξνL=1(j)=∑n=2,3,4,…ξνL=1(j;n),
where ξνL=1(j;n) indicates the nth order of the perturbation energy. Substitution of (74) and (75) into (73) yields
(76)ET=W+Epair≈C(L1,…,LN;p1,…,pN)+Nλ+∑i=1NU(pi(eB))+12(ℏeBm*)NνL=0+32(ℏeBm*)NνL=1+NνL=1×∑j=1,2,3,…(∑n=2,3,4,…ξνL=1(j;n))inthegroundstatewith2<ν<3.
Therein the function form of W is continuous with the change in ν. On the other hand, the pair energy Epair has a discontinuous form for the argument ν, because the number of the allowed transitions depends discontinuously upon ν. This discontinuous property produces the plateaus of the Hall resistance at specific filling factors. We have already calculated the second order perturbation energies for j=1 and 2 as in (21), (31), (37), (41b), (45), (47), and (49). We list the results in Tables 2 and 3.

Table 2

Second order of the perturbation energy per electron for the electron pairs placed in the second nearest Landau orbital pairs.

ν

5/2

48/19

78/31

8/3

14/5

18/7

19/7

ξνL=1(2;2)

-(1/2)S

-(41/190)S

-(113/496)S

-S/6

-S/20

-3S/28

-2S/35

Table 3

Second order of the perturbation energy per electron for the electron pairs placed in the nearest Landau orbital pairs.

ν

5/2

48/19

78/31

8/3

14/5

18/7

19/7

ξνL=1(1;2)

0

-(9/190)T

-(15/496)T

-T/6

-T/20

-3T/28

-2T/35

Now we examine the effects of further neighbouring electron pairs in the ν=14/5 state. Figure 20 shows the most uniform configuration. Therein the allowed transitions from the electron pairs AnBn(n=1,2,3,…) are shown by the arrow pairs. The centre position between the nearest pair A1B1 is equal to that of the electron pair AnBn for any integer n>1. Accordingly, the total momentum of the pair AnBn is equal to that of the pair A1B1. Therefore, the electron pair AnBn with L=1 can transfer to all the empty states as the pair A1B1.

Figure 20

Various electron pairs with the same total momentum at ν=14/5. Dashed lines indicate empty orbitals and solid lines indicate filled orbitals in the Landau level L=1. Allowed transitions from the electrons A1 and B1 are shown by black arrow pairs, from A2B2 by blue, from A3B3 by brown, and from A4B4 by dark green.

Also, the total momentum of the electron pair C1D1 in Figure 21 is equal to that of the pairs CnDn(n=1,2,3,…) and therefore the pair CnDn can transfer to all the empty states except the orbital Y shown in blue. This only one forbidden transition may be ignored in comparison with the enormously many allowed transitions which are caused by the spreading of the Landau wave function in the y-direction for the macroscopic size of the device.

Figure 21

Various electron pairs with the same total momentum at ν=14/5. Dashed lines indicate empty orbitals and solid lines indicate filled orbitals in the Landau level L=1. Allowed transitions from the electrons C1 and D1 are shown by black arrow pairs, from C2D2 by blue, from C3D3 by brown, and from C4D4 by dark green.

Thus, the further neighbouring electron (or hole) pairs with j≥3 can transfer to all the empty (or filled) orbitals at ν=8/3,14/5,7/3, and 11/5. The energies of these pairs with j≥3 are negative in the second order perturbation. Therefore, the energies are accumulated to give a stronger binding energy and so the states become more stable.

8. Conclusions

The FQH states with 2<ν<3 have been investigated by using the method developed in previous articles [25–32]. We have found the most uniform configuration in the Landau orbitals at ν=5/2,48/19,78/31,8/3,14/5,18/7,19/7,7/3,11/5,17/7, and 16/7. Especially, the electron (or hole) pairs placed in the first and second neighbouring Landau orbitals in L=1 can transfer to all the empty orbitals at ν0=8/3,14/5,7/3, and 11/5 states via the Coulomb interaction. Also, at ν0=5/2, the electron pairs placed in the second nearest Landau orbitals can transfer to all the empty orbitals with L=1. More distant electron (or hole) pairs with the same centre positions as in the first and second nearest pairs can also transfer to all the empty (or filled) orbitals at ν0. Then, the energies of the distant pairs, ξνL=1(j) for j≥3, are accumulated to that of the first and second nearest pairs.

This capability of the quantum transitions to all the empty orbitals means that the number of transitions is largest at ν0. Accordingly, the number of transitions decreases abruptly when the filling factor deviates from ν0=5/2,8/3,14/5,7/3, and 11/5. This property is caused by the combined effect of the most uniform configuration, momentum conservation and Fermi-Dirac statistics. For example, (41a), (41b)–(43) show that the pair energy at ν=8/3 becomes half of that in the neighbourhood of ν=8/3. The spectrum of the pair energy has a valley structure at ν0=5/2,8/3,14/5,7/3,11/5 and so on. That is to say, the pair energy has a discontinuous function form which takes the lowest value at the specific filling factor ν0 and becomes higher energy with a gap in the neighbourhood of ν0. Therefore, the ν0-FQH states are stable at ν0= 5/2, 8/3, 14/5, 7/3, and 11/5. Since thousands of the Landau wave functions are overlapping with each other, the deviation of the Hall resistance from h/(e2ν0) becomes smaller than 0.1%. This property is in agreement with the experimental value, the accuracy of which is 0.015% at ν0=7/2 and so on. Thus, we should study the quasi-2D system with more than thousand electrons. Our treatment can do this task because the present theory can count the number of transitions for an enormous number of electrons.

When we choose the parameter-ratio (T/S)=2, the theoretical ratio of the pair energies at ν0=5/2,8/3,14/5, and 19/7 is equal to 1 : 1 : (3/10) : (12/35) which is in reasonable agreement with the data of the high mobility sample in [9].

The present theory has explained the FQH phenomena for various filling factors 2<ν<3 based on a standard treatment of interacting quasi-2D electron gas without assuming any quasiparticle.

Acknowledgments

The author expresses his heartfelt appreciation for an important suggestion of Professor Klaus von Klitzing. He has suggested to the author to examine the 5/2 plateau problem. Also the author wishes to express his appreciation for the encouragement of Professor Koichi Katsumata, Professor Masayuki Hagiwara, Professor Hidenobu Hori, and Professor Yasuyuki Kitano. Particularly Professor Katsumata has given his important suggestions for improving my description. The author could not have completed this paper without their support.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.